Solve Vector Potential Formulation of Navier-Stokes Equations: Using Fourth-Order-Accuracy Localized Differential Quadrature Method
Date Issued
2010
Date
2010
Author(s)
Tsai, Chia-Hsing
Abstract
To simulate three-dimensional flow problems in this thesis, the vector potential formulations of three-dimensional incompressible Navier-Stokes are chosen to govern the motion of fluid flow. The vector potential formulation belongs to one of the pressure-free algorithms which are obtained by taking curl to the momentum equations. By replacing the vorticity with Laplacian vector potential to the vorticity transport equations, the Navier-Stokes equations are transformed to fourth-order partial differential equations (PDEs) with one variable---vector potential. Comparing with other pressure-free algorithms, vector potential formulations are simpler and more accurate, and, moreover, the computation is more efficient. To the boundary conditions of vector potential, except the presented defined boundary conditions for confined flow, we further improved the algorithm to through-flow problem by introducing the concept of Stokes'' theorem. To author''s best knowledge, this improvement is groundbreaking. To accurately approximate these fourth-order governing equations, fourth-order-accuracy localized differential quadrature (LDQ) methods are employed. Through adopting the non-uniform mesh grids, the solutions can be obtained efficiently. To examine the ability of the proposed scheme to fourth-order governing equations, two benchmark problems are considered, including two-dimensional cavity flow problems and backward-facing step flow problems. The results show the accuracy and feasibility of the proposed scheme. By the successful implementation of the present scheme to two-dimensional flow problems, the proposed scheme is further employed to solve three-dimensional benchmark problems, including three-dimensional driven cavity flow problems and backward-facing step flow problems. The good performance not only demonstrates that the proposed scheme is able to be employed to solve the vector potential formulation, but also validates the correctness of the presented formulation. Furthermore, we specifically visualized the contour of vector potential by numerical simulation. The comparison between vector potential and stream functions show the difference of these two algorithms. Conclusively, the vector potential formulations of Navier-Stokes equations are successfully used to simulate the three-dimensional fluid motion, especially the fluid flow problems with through-flow. Through the application of the fourth-order-accuracy of localized differential quadrature method, the solutions can be accurately obtained, and the vector potential can be specifically visualized. From the previous literatures, no literature has ever presented the similar idea of this research. It is convinced that the groundbreaking findings in this thesis can provide a feasible way to simulate three-dimensional fluid motion.
Subjects
Localized differential quadrature method
vector potential formulation
stream function formulation
Navier-Stokes equations
fourth-order partial differential equations
Type
thesis
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